On Distributionally Robust Chance Constrained Program with Wasserstein Distance Weijun Xie ISE, Virginia Tech Mathematical Optimization of Systems Impacted by Rare, High-Impact Random Events, Jun 24 - 28, 2019
Distributionally Robust Chance Constrained Program (DRCCP) Consider DRCCP as v ∗ = c ⊤ x min (objective function) x (deterministic constraints) x ∈ S s.t. e.g., nonnegativity Ax ≥ ˜ ˜ b (uncertain inequalities) Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 2 / 27
Distributionally Robust Chance Constrained Program (DRCCP) Consider DRCCP as v ∗ = c ⊤ x min (objective function) x (deterministic constraints) x ∈ S s.t. e.g., nonnegativity P ∈P P { ˜ Ax ≥ ˜ inf b } ≥ 1 − ǫ (chance constraint) where ◮ ǫ ∈ (0 , 1) is risk parameter ◮ “Ambiguity Set” P = a family of probability distributions Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 2 / 27
Distributionally Robust Chance Constrained Program (DRCCP) Consider DRCCP as v ∗ = c ⊤ x min (objective function) x (deterministic constraints) x ∈ S s.t. e.g., nonnegativity 1 x ≥ ˜ a ⊤ ˜ b 1 . . inf ≥ 1 − ǫ P ∈P P (chance constraint) . m x ≥ ˜ a ⊤ ˜ b m where ◮ ǫ ∈ (0 , 1) is risk parameter ◮ “Ambiguity Set” P = a family of probability distributions ◮ m = 1 : single DRCCP; m > 1 : joint DRCCP Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 2 / 27
Wasserstein Ambiguity Set Wasserstein ambiguity set (Esfahani and Kuhn 2015; Zhao and Guan, 2015; Gao and Kleywegt, 2016; Blanchet and Murthy, 2016) � � � � P W = P : W q P , P ˜ ≤ δ , ζ � � where W q P , P ˜ = Wasserstein distance between probability distribution P ζ and empirical distribution P ˜ ζ . Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 3 / 27
Wasserstein Ambiguity Set Wasserstein ambiguity set (Esfahani and Kuhn 2015; Zhao and Guan, 2015; Gao and Kleywegt, 2016; Blanchet and Murthy, 2016) � � � � P W = P : W q P , P ˜ ≤ δ , ζ � � where W q P , P ˜ = Wasserstein distance between probability distribution P ζ and empirical distribution P ˜ ζ . ◮ Convergence in probability to regular chance constrained program (CCP) ◮ Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 3 / 27
DRCCP with Wasserstein Ambiguity Set (DRCCP-W): Existing Works DRCCP-W set � � � ˜ � Ax ≥ ˜ Z = x : inf ≥ 1 − ǫ , P ∈P W P b � � � � with P W = P : W q P , P ˜ ≤ δ . ζ ◮ Hanasusanto et al. (2015) and X. and Ahmed (2017) showed that DRCCP-W is a biconvex program. Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 4 / 27
DRCCP with Wasserstein Ambiguity Set (DRCCP-W): Existing Works DRCCP-W set � � � ˜ � Ax ≥ ˜ Z = x : inf ≥ 1 − ǫ , P ∈P W P b � � � � with P W = P : W q P , P ˜ ≤ δ . ζ ◮ Hanasusanto et al. (2015) and X. and Ahmed (2017) showed that DRCCP-W is a biconvex program. ◮ X. and Ahmed (2017) proposed a bicriteria approximation algorithm for a special family of DRCCP-W Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 4 / 27
DRCCP-W: Summary of Contributions DRCCP-W set � � � ˜ � Ax ≥ ˜ Z = x : inf ≥ 1 − ǫ . P ∈P W P b ◮ DRCCP-W ≡ conditional-value-at-risk (CVaR) constrained optimization � Develop inner and outer approximations ◮ DRCCP-W set Z is mixed integer program representable � With big-M coefficients and additional binary variables ◮ Binary DRCCP-W set (i.e., S ⊆ { 0 , 1 } n ) is submodular constrained � Without big-M coefficients and additional binary variables � Solvable by Branch and Cut Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 5 / 27
Outline ◮ CVaR Reformulation and Related Approximations ◮ Mixed Integer Program Reformulation ◮ Binary DRCCP-W and Submodularity ◮ Concluding Remarks Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 6 / 27
CVaR Reformulation and Related Approximations Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 7 / 27
CVaR Reformulation DRCCP-W set � � � ˜ � Ax ≥ ˜ Z = x : inf P ∈P W P b ≥ 1 − ǫ , � � � � with P W = P : W q P , P ˜ ≤ δ . ζ Theorem (Exact Formulation) � � � � x : δ − f ( x, ˜ Z = ǫ + CVaR 1 − ǫ ζ ) ≤ 0 , � ( a i , b i ) − ( a ζ i , b ζ where f ( x, ζ ) := min inf i ) � and i ∈ [ m ] a ⊤ i x<b i � � � � � � γ + 1 ˜ ˜ X = min X − γ CVaR 1 − ǫ ǫ E P . γ + Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 8 / 27
CVaR Reformulation DRCCP-W set � � � ˜ � Ax ≥ ˜ Z = x : inf P ∈P W P b ≥ 1 − ǫ , � � � � with P W = P : W q P , P ˜ ≤ δ . ζ Theorem (Exact Formulation) � � � � x : δ − f ( x, ˜ Z = ǫ + CVaR 1 − ǫ ζ ) ≤ 0 , � ( a i , b i ) − ( a ζ i , b ζ where f ( x, ζ ) := min inf i ) � and i ∈ [ m ] a ⊤ i x<b i � � � � � � γ + 1 ˜ ˜ X = min X − γ CVaR 1 − ǫ ǫ E P . γ + Proof Idea: (1) strong duality of distributionally robust optimization, and (2) break down the indicator function. Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 8 / 27
CVaR Reformulation: Worst-case Interpretation Theorem (Exact Formulation) � � � � x : δ − f ( x, ˜ Z = ǫ + CVaR 1 − ǫ ζ ) ≤ 0 , � ( a i , b i ) − ( a ζ i , b ζ where f ( x, ζ ) := min inf i ) � i ∈ [ m ] a ⊤ i x<b i Original empirical samples ◮ N = 6 , ǫ = 1 / 3 Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 9 / 27
CVaR Reformulation: Worst-case Interpretation Theorem (Exact Formulation) � � � � x : δ − f ( x, ˜ Z = ǫ + CVaR 1 − ǫ ζ ) ≤ 0 , � ( a i , b i ) − ( a ζ i , b ζ where f ( x, ζ ) := min inf i ) � i ∈ [ m ] a ⊤ i x<b i Original empirical samples Moving these samples to boundary of violating constraints ◮ N = 6 , ǫ = 1 / 3 Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 9 / 27
CVaR Reformulation: Worst-case Interpretation Theorem (Exact Formulation) � � � � x : δ − f ( x, ˜ Z = ǫ + CVaR 1 − ǫ ζ ) ≤ 0 , � ( a i , b i ) − ( a ζ i , b ζ where f ( x, ζ ) := min inf i ) � i ∈ [ m ] a ⊤ i x<b i Original empirical samples Moving these samples to Due to chance constraint, boundary of violating only limited scenarios can constraints be moved ◮ N = 6 , ǫ = 1 / 3 Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 9 / 27
CVaR Reformulation: Simplification DRCCP-W set � � � � x : δ − f ( x, ˜ Z = ǫ + CVaR 1 − ǫ ζ ) ≤ 0 , � ( a i , b i ) − ( a ζ i , b ζ where f ( x, ζ ) := min inf i ) � . i ∈ [ m ] a ⊤ i x<b i Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 10 / 27
CVaR Reformulation: Simplification DRCCP-W set � � � � x : δ − f ( x, ˜ Z = ǫ + CVaR 1 − ǫ ζ ) ≤ 0 , � � ( a ζ i ) ⊤ x − b ζ max i , 0 where f ( x, ζ ) := min . � ( x, 1) � ∗ i ∈ [ m ] Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 10 / 27
CVaR Reformulation: Simplification DRCCP-W set � � � � x : δ − � f ( x, ˜ Z = ǫ � ( x, 1) � ∗ + CVaR 1 − ǫ ζ ) ≤ 0 , � � where � ( a ζ i ) ⊤ x − b ζ f ( x, ζ ) := min i ∈ [ m ] max i , 0 . ◮ By positive homogeneity of coherent risk measures Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 10 / 27
CVaR Reformulation: Simplification DRCCP-W set � � � � x : δ − � f ( x, ˜ Z = ǫ � ( x, 1) � ∗ + CVaR 1 − ǫ ζ ) ≤ 0 , � � � � where � ( a ζ i ) ⊤ x − b ζ f ( x, ζ ) := max min , 0 . i i ∈ [ m ] ◮ By positive homogeneity of coherent risk measures ◮ Switch minimax to maximin Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 10 / 27
Outer Approximation Note � � � � � � ˜ ˜ X ≥ VaR 1 − ǫ X := min s : F ˜ X ( s ) ≥ 1 − ǫ . CVaR 1 − ǫ � � � � ˜ ˜ X X Replace CVaR 1 − ǫ by VaR 1 − ǫ . Theorem (Outer Approximation) � � � � x : δ − � f ( x, ˜ Z = ǫ � ( x, 1) � ∗ + CVaR 1 − ǫ ζ ) ≤ 0 � � � � where � ( a ζ i ) ⊤ x − b ζ f ( x, ζ ) := max min , 0 . i i ∈ [ m ] Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 11 / 27
Outer Approximation Note � � � � � � ˜ ˜ X ≥ VaR 1 − ǫ X := min s : F ˜ X ( s ) ≥ 1 − ǫ . CVaR 1 − ǫ � � � � ˜ ˜ X X Replace CVaR 1 − ǫ by VaR 1 − ǫ . Theorem (Outer Approximation) � � � � x : δ − � f ( x, ˜ Z ⊆ Z VaR = ǫ � ( x, 1) � ∗ + VaR 1 − ǫ ζ ) ≤ 0 � � � � where � ( a ζ i ) ⊤ x − b ζ f ( x, ζ ) := max min , 0 . i i ∈ [ m ] Xie (Virginia Tech) DRCCP with Wasserstein Distance June 26, 2019 11 / 27
Recommend
More recommend
